{smcl}
{* 07jul2006}{...}
{cmd:help mata mm_root()}
{hline}

{title:Title}

{p 4 19 2}
{bf:mm_root() -- Brent's univariate zero (root) finder}


{title:Syntax}

{p 8 21 2}{it:rc} = {cmd:mm_root(}{it:x}{cmd:,} {it:f}{cmd:,}
{it:lo}{cmd:,} {it:up} [{cmd:,} {it:tol}{cmd:,} {it:itr}{cmd:,}
{it:...}]{cmd:)}

{pstd}
where

{p 11 16 2}
{it:rc}:  the return code; {it:rc}!=0 indicates that
no valid solution has been found

{p 12 16 2}
{it:x}:  will be replaced by a {it:real scalar} containing solution

{p 12 16 2}
{it:f}:  {it:pointer scalar} containing address of function whose zero
will be sought for; usually
this is coded {cmd:&}{it:funcname}{cmd:()}

{p 11 16 2}
{it:lo}:  {it:real scalar} containing lower endpoint of the search interval

{p 11 16 2}
{it:up}:  {it:real scalar} containing upper endpoint of the search interval

{p 10 16 2}
{it:tol}:  {it:real scalar} specifying acceptable tolerance for the root
estimate (default is {it:tol}=0 to find the root as
accurate as possible)

{p 10 16 2}
{it:itr}:  {it:real scalar} specifying the maximum number of
iterations (default is {it:itr}=1000)

{p 10 16 2}
{it:...}:  up to 10 additional arguments to pass on to function {it:f}


{title:Description}

{pstd}{cmd:mm_root()} searches the interval from {it:lo} to {it:up}
for the root of function {it:f} with respect to its first argument.
That is, {cmd:mm_root()} approximates the value {it:x} for which
{it:f}{cmd:(}{it:x} [{cmd:,} {it:...}]{cmd:)} evaluates to zero. The
accuracy of the approximation is 4*epsilon({it:x}) + {it:tol}.

{pstd}{cmd:mm_root()} stores the found solution in {it:x} and issues
return code {it:rc}. Possible return codes are:

{phang2}{bind: 0:} everything went well

{phang2}{bind: 1:} convergence has not been reached within the
maximum number of iterations; {it:x} will contain the current approximation

{phang2}{bind: 2:} {it:f}{cmd:(}{it:lo}{cmd:)} and {it:f}{cmd:(}{it:up}{cmd:)} do not
have opposite signs and {it:f}{cmd:(}{it:lo}{cmd:)} is closer to
zero than {it:f}{cmd:(}{it:up}{cmd:)}; {it:x} will be set to {it:lo}

{phang2}{bind: 3:} {it:f}{cmd:(}{it:lo}{cmd:)} and {it:f}{cmd:(}{it:up}{cmd:)} do not
have opposite signs and {it:f}{cmd:(}{it:up}{cmd:)}  is closer to
zero than {it:f}{cmd:(}{it:lo}{cmd:)}; {it:x} will be set to {it:up}

{pstd}
{cmd:mm_root()} is a (slightly modified) translation of the C
realization of Brent's zero finder provided in
{browse "http://www.netlib.org/c/brent.shar"}. A description of the
algorithm and details on the modifications can be found in the source
of {cmd:mm_root()} (see below).


{title:Remarks}

{pstd}Example:

        {com}: function myfunc(x, a) return(x^2 - a)

        : a = 2/3
        {res}
        {com}: mm_root(x=., &myfunc(), 0, 1, 0, 1000, a)
        {res}  0

        {com}: x
        {res}  .8164965809

        {com}: mm_root(x=., &myfunc(), 0, 1, 0.01, 1000, a)
        {res}  0

        {com}: x
        {res}  .8168350168

        {com}: sqrt(a)
        {res}  .8164965809{txt}


{title:Conformability}

{pstd}
{cmd:mm_root(}{it:x}{cmd:,} {it:f}{cmd:,} {it:lo}{cmd:,} {it:up}{cmd:,}
{it:tol}{cmd:,} {it:itr}{cmd:,} {it:...}{cmd:)}:{p_end}
           {it:x}:  input: anything; output: 1 {it:x} 1
           {it:f}:  1 {it:x} 1
          {it:lo}:  1 {it:x} 1
          {it:up}:  1 {it:x} 1
         {it:tol}:  1 {it:x} 1
         {it:itr}:  1 {it:x} 1
         {it:...}:  (depending on function {it:f})
      {it:result}:  1 {it:x} 1


{title:Diagnostics}

{pstd}{it:x} will be set to missing if {it:f} evaluates to
missing at some point in the algorithm.


{title:Source code}

{pstd}
{help moremata_source##mm_root:mm_root.mata}


{title:Author}

{pstd} Ben Jann, University of Bern, jann@soz.unibe.ch


{title:Also see}

{psee}
Online:  help for
{bf:{help mf_mm_nrroot:mm_nrroot()}},
{bf:{help m2_ftof:[M-2] ftof}},
{bf:{help moremata}}
